ECON 4910 seminar 21 ECON 4910 Spring 2007 Second seminar Lecturer Finn R. Førsund

ECON 4910 seminar 2 2 The Meade externality models Consider the following externality models found in Meade (1952): a) Unpaid factor b) Reciprocal c) Atmosphere

ECON 4910 seminar 2 3 a) Unpaid factor Output from one activity creates an externality in another production activity, using the same type of input, example is an apple orchard and a honey producer located together Discuss the sign of

ECON 4910 seminar 2 4 Problem formulation for Pareto optimality Problem formulation

ECON 4910 seminar 2 5 Solving the problem The Lagrangian (eliminating y 1, y 2 )

ECON 4910 seminar 2 6 Solving the problem, cont. First-order conditions Assuming interior solutions for x 1,x 2 Marginal benefit (creating objective function value) equal to marginal cost (opportunity cost of using the input)

ECON 4910 seminar 2 7 Solving the problem, cont. 1 Interpretation of the shadow prices  The Envelope Theorem  λ: Increase in y 1 when y 2 decreases with one unit  γ: Increase in y 1 when x increases with one unit

ECON 4910 seminar 2 8 Solving the problem, cont. 2 Solution for 4 endogenous variables x 1,x 2, λ,γ: 2 first-order conditions plus 2 constraints Forming the ratio of productivities Value of ratio of marginal productivities different from the trade-off λ between y 1 and y 2

ECON 4910 seminar 2 9 Competitive market solution Maximising profit taking output- and input prices p,q as given

ECON 4910 seminar 2 10 Competitive market solution, cont. First-order conditions Ratio of marginal productivities equal to trade-off between outputs neglects the externality and implies too little of the resource used in activity 1 and too much in activity 2 when the externality is positive

ECON 4910 seminar 2 11 Demonstration: comparing Pareto optimum with market solution Pareto optimum λ is the trade-off between outputs Assuming prices fixed λ = p 2 /p 1  Market solution p 2 /p 1 is the market trade-off between outputs

ECON 4910 seminar 2 12 Implementing the Pareto-optimal solution Introducing a Pigou tax (subsidy) on the activity generating an externality Profit maximising problem with subsidy

ECON 4910 seminar 2 13 Implementing the Pareto-optimal solution, cont. The market solution with a Pigou subsidy Finding the optimal subsidy rate

ECON 4910 seminar 2 14 Policy discussion Alternative: merging the firms The optimisation problem of the merged firm Solution

ECON 4910 seminar 2 15 The case of reciprocal externality Reciprocal externalities imply that the relative strength of the externalities determines the resource allocation Setting up the optimisation problem for finding the Pareto solution we cannot eliminate the outputs y 1, y 2 An apparently innocent change makes the mathematics much more cumbersome Two subsidies may be used in principle

ECON 4910 seminar 2 16 The atmosphere externality case The Lagrangian for the Pareto problem First-order conditions

ECON 4910 seminar 2 17 The atmosphere externality case, cont. Forming the rate of productivities The competitive market solution

ECON 4910 seminar 2 18 Implementing the Pareto-optimal solution Introducing a Pigou subsidy on the externality-generating output from activity 1 The market solution with a subsidy Finding the optimal subsidy rate setting market solution equal to optimal solution inserting for λ